In academic research paper all sections are linked:
Introduction ➡️ Literature Review ➡️ Method ➡️ Results ➡️ Discussion & Conclusion
To understand the statistics in the results section it is essential to identify the concepts presented in each section:
Variables
A variable …
- Is way of assigning values (numbers or characters) to labels
- Corresponds to a column in a spreadsheet
Challenge: Identify the Role and the Type of each variable
Type of Variables
There are many type for variables but the only that we should care are … - Continuous: If values are numbers - Categorical: If values are characters
Note: Distinguish Categorical Nominal variables (e.g., Irish, French) vs. Categorical Ordinal variables (e.g., XS, S, M, L, XL)
Role of Variables
A variable can have one or the other of these roles (no other role exist):
- Outcome: “to be explained” variable as Y (also called Dependent Variable or DV)
- Predictor: “doing the explaining” as X (also called Independent Variable or IV)
Note: A variable can be also both but in different hypotheses
Hypotheses
Main Effect Hypothesis Formulation
The Outcome has to be Continuous but …
- Case 1: Predictor is Continuous
The {outcome} increases when {predictor} {increases/decreases/changes}
- Case 2: Predictor is Categorical (2 Categories)
The {outcome} of {predictor category 1} is {higher/lower/different} than the {outcome} of {predictor category 2}
- Case 3: Predictor is Categorical (3 or more Categories)
The {outcome} of at least one {predictor} category is {higher/lower/different} than the other {predictor} categories
Model & Equation
The basic structure of a statistical model is:
\[Outcome = Model + Error\]
where the \(Model\) is a series of predictors that are expressed in hypotheses related to the same outcome. - Main effect hypotheses are indicated with the predictor name only - Interaction effect hypotheses are indicated with all predictor names separated by \(*\)
Example:
\[Outcome = Pred1 + Pred2 + Pred1 * Pred2 + Error\]
To evaluate their relationship with the outcome, each effect hypothesis is related with a coefficient called Estimate and represented with \(\beta\) as follow:
\[Outcome = \beta_0 + \beta_1 Pred1 + \beta_2 Pred2 + \beta_3 Pred1 * Pred2 + Error\]
Note: \(\beta_0\) is the estimate related to the intercept. It is always included, always tested but has no interest in the analysis
Evaluation of the Significance
Testing for the significance of the effect means evaluating if this estimate \(\beta\) value is significantly different, higher or lower than 0 as hypothesised in \(H_a\):
- \(\beta \neq 0\) means our hypothesis doesn’t precise the direction of the change, just that there is a change
- \(\beta > 0\) means our hypothesis indicates that the relationship increases or a group is higher than another group
- \(\beta < 0\) means our hypothesis indicates that the relationship decreases or a group is lower than another group
Note: \(H_0\) will always predict that \(\beta = 0\)
The significance, called \(p\)-value, is the probability to consider \(H_0\) as True. This probability is between 0% and 100% which corresponds to a value between 0.0 and 1.0.
If the \(p\)-value:
- Is higher than 5% or 0.05, then \(H_0\) is accepted
- Is lower than 5% or 0.05, then \(H_0\) is rejected and \(H_a\) is considered as plausible
Graphic Representation of a Model
A graphic representation of the model’s hypothesised effects can be done: - All the arrows correspond to an hypothesis to be tested - All the tested hypotheses have to be represented with an arrow
A simple arrow is a main effect
A crossing arrow is an interaction effect
Note: By default, an interaction effect involves the test of the main effect hypotheses of all Predictors involved
Statistical Test
JAMOVI: Stats. Open. Now.
Jamovi an be downloaded or used online on https://www.jamovi.org/
A free book “Learning Statistics with Jamovi” by Navarro and Foxcroft (2019) is available online here: https://www.learnstatswithjamovi.com/
Advantages:
- Free
- Simple Interface
- No Missing Values to Declare
- No Variable to Recode by Default
- Ready to Publish Tables and Figures
- Free Modules for Advanced Statistics (Mediation, Generalized LM, Linear Mixed Model)
Note: In Jamovi …
- The outcome is called Dependent Variable
- A continuous predictor is a covariate
- A categorical predictor is a factor
Hypotheses with Continuous Predictors and with Categorical Predictors Having 2 Categories
Steps:
- Open your file
- Check the type of your variables
- Analyses > Regression > Linear Regression
- Set the Outcome as DV and
- To test the main effect hypotheses: set the Predictors as Covariates/Factors
- To test interaction effect hypotheses: In Model Builder, select all predictor with
CTRL (win) or Command (mac) and bring them as interaction in the model
Communicate the Results about the full model and each hypothesis:
- Use Model Fit Measure Table to evaluate the accuracy of the full model
The predictions from a model including all effects are significant/not-significant better than without these effects ( \(R^2 = value_{R^2}\), \(F(df1,df2) = value_{F}\), \(p = value_{p}\))
- Use Model Coefficients Table to conclude about each hypothesis
The effect of \(Predictor\) on \(Outcome\) is statistically significant/not-significant, therefore \(H_0\) can be rejected/accepted ( \(b = value_{estimate}, 95\% CI [lower\,CI, upper\,CI]\), \(t(df) = value_t\), \(p = value_{p}\)).
Hypotheses with Categorical Predictors Having 3 or more Categories
- Open your file
- Check the type of your variables
- Analyses > Regression > Linear Regression
- Set the Outcome as DV and
- To test the main effect hypotheses: set the Predictors as Factors
- To test interaction effect hypotheses: In Model Builder options, select all predictor with
CTRL (win) or Command (mac) and bring them as interaction in the model
- Tick ANOVA Test in Model Coefficient options
Communicate the Results about the full model and each hypothesis:
- Use Model Fit Measure Table to evaluate the accuracy of the full model
The predictions from a model including all effects are significant/not-significant better than without these effects ( \(R^2 = value_{R^2}\), \(F(df1,df2) = value_{F}\), \(p = value_{p}\))
- Use Omnibus ANOVA Test Table to conclude about each hypothesis
The effect of \(Predictor\) on \(Outcome\) is statistically significant/not-significant, therefore \(H_0\) can be rejected/accepted ( \(F(df_{predictor}, df_{residual}) = value_F\), \(p = value_{p}\)).
Discussion & Conclusion
From here…
- There is no number to be shown and no specific guidelines
- Correct interpretation comes if results have been understood and if reasons for the results to be the ones obtained have been identified
